Cone in toric geometry

A cone in toric geometry is a convex cone generated by finitely many lattice vectors and containing no nonzero linear subspace. Such cones are the pieces of a fan in toric geometry.

A cone in toric geometry is the nonnegative real span of finitely many lattice vectors. It is rational when those generators lie in a lattice and strictly convex when it contains no nonzero linear subspace.

A face of a polyhedral cone $C$ is $C\cap \ker \ell$ for a linear functional nonnegative on $C$. Every convex face of a polyhedral cone is exposed in this way. Faces include the whole cone and its lineality space, and are closed under intersection.

// Target: geometry-and-topology.bigb

A ray of a fan is a one-dimensional cone in a fan in toric geometry. It has the form ${\mathbb{R}}_{\ge 0}v$, where $v$ is a primitive lattice vector, and occurs as a face of a polyhedral cone.

A ray of a fan is a one-dimensional cone. It has a unique primitive lattice generator $v_{\rho }$ pointing along it.

For a cone $\sigma \subseteq N_{\mathbb{R}}$, its dual cone is

For a closed polyhedral cone $C$, the maps $\tau \mapsto C^{\vee }\cap {\tau }^{\perp }$ and $\nu \mapsto C\cap {\nu }^{\perp }$ give inverse order-reversing bijections on faces. Choose a exposing functional in the relative interior of the dual face. The bipolar identity and an exposing functional for the original face show that applying the maps twice returns that face.

// Target: geometry-and-topology.bigb

For a rational cone $\sigma$, the lattice points ${\sigma }^{\vee }\cap M$ form a finitely generated additive monoid.

The lattice points of a rational polyhedral cone form a finitely generated additive monoid. Choose lattice generators for the cone. Subtracting integer parts of nonnegative coefficients leaves a lattice point in their bounded fundamental parallelepiped; its finitely many lattice points, together with the chosen generators, generate the monoid. This supplies finite generation of the coordinate ring of an affine toric variety.

// Target: geometry-and-topology.bigb

The Hilbert basis is the unique minimal generating set of the additive monoid of lattice points in a pointed rational cone.

A polyhedral cone is simplicial when its primitive ray generators are linearly independent. A smooth cone in a lattice must be simplicial, but simpliciality alone does not make its ray generators part of a lattice basis.